I will preface my opinion with two things: we have three young children and travel pretty frequently (i.e. I get it and the hassle of all the baby gear) and this isn't the most popular option based on my observations. I highly recommend purchasing an airline seat for your child. It is safer for both him and the car seat. Even gate checking a car seat isn't recommended. My littles actually do really well in their seats as they are used to being strapped in them. It is easier for them to nap. And it gives everyone their own (albeit still very small) space. No, nobody is safe in a catastrophic crash, but a properly secured baby is much safer in every other air travel situation.
Ok, so now on to your questions. Definitely use your own car seat in the shuttles. You don't have any idea the history or safety of rented/borrowed car seats nor if it is properly installed or what other child has pooped through, vomitted, etc. and if that happened if the seat was properly cleaned for you and for the integrity of the seat. Car seats expire and I have seen so many car seats that look well past their expiration sitting behind the shuttle check-in counter or thrown in the bottom of the luggage compartment.
When we were at the parks, we left our diaper bag attached to the stroller when we went on rides. I wore a cross-body bag with a small water bottle and few little snacks that the kids could have while in line, though we did avoid anything with more than a 15-20 min wait. I also carried our park tickets, credit cards, phone. We used our phone for our camera so we didn't have to deal with a separate one.
I think you and your DH should take turns riding a few of your favorites. Mid-morning the Radiator Springs single rider line is fast. We did that and both of us had a <10 minute wait. I'm not the expert on single rider lines vs rider swap, but definitely make use of them.
My biggest tip on traveling with littles to any destination, Disney included, are sleep and food. Make sure your little one has enough chances to rest. Even if it means you and your husband take turns taking him back to the hotel for one good nap a day and attempting to get him to bed close to his normal time. And pack a small soft sided cooler with plenty of snacks. I swear that a little snack cup with some gold fish and cut up fruit, mini boxes of raisins, apple sauce pouch, etc can do wonders!
Not sure when you are going, but the Princess Dot Puddle Park was a HIT with my littles. Bring an extra outfit! And have so much fun!